Compounded Rates

In order to reduce the impact of outliers in daily fluctuations of financial markets, many financial products use simple or compounded averages of a given reference rate. For instance, on every business day the New York Federal Reserve Bank publishes the average rate of the Secured Overnight Financing Rate, compounded over the last 30 calendar days under the name of SOFR30.
Standard Methodology
Here, we present the methodology of compounded rates that is used by large banks such as the New York Federal Reserve Bank (compounded SOFR) and the Bank of England (compounded SONIA).
Consider a unit investment over a time period of dcd_cdc​
 calendar days containing db≤dcd_b\leq d_cdb​≤dc​
 business days. Let rir_iri​
 be an interest rate that is published once every business day and assume that the business day convention is such that the year has NNN
 days. Then, the compounded gain GGG
 over the whole interest period results to
​
 .
We remark that for constant interest rates ri=rr_i = rri​=r
 and ni=1n_i = 1ni​=1
 this simplifies to the well-known compounded gain formula G=(1+rN)dbG=\left(1+\frac{r}{N}\right)^{d_b}G=(1+Nr​)db​
.
However, in general, the rate factor nin_ini​
 is an integer value that accounts for iii
 being a business day or a non-business day. More precisely, if iii
 is a business day followed by kkk
 non-business days, we set ni=k+1n_i = k+1ni​=k+1
. For instance, if iii
 is followed by a business day, i.e., k=0k=0k=0
, we have ni=1n_i=1ni​=1
. For a friday, which is usually followed by two non-business days, we would have ni=3n_i = 3ni​=3
.
Now, in order to get the average interest III
 from the compounded gain GGG
, we subtract the original investment and normalize, thus obtaining
​I=Ndc[∏i=1db(1+ri×niN)−1]I = \frac{N}{d_c}\left[\prod_{i=1}^{d_b}\left( 1 + \frac{r_i \times n_i}{N} \right) -1\right]I=dc​N​[∏i=1db​​(1+Nri​×ni​​)−1]
​
which is the formulation of compounded rates used by the FED and the BOE amongst others.
Sources:
https://www.newyorkfed.org/medialibrary/Microsites/arrc/files/2019/Users_Guide_to_SOFR.pdf

https://www.bankofengland.co.uk/paper/2020/supporting-risk-free-rate-transition-through-the-provision-of-compounded-sonia​
DIA Methodology
The methodology from the previous section has a special feature in that it mixes compounded and non-compounded rates. More precisely, investments are not compounded for weekends and holidays. This behaviour is reflected in the rate factor nin_ini​
. In the Index IDIAI_{DIA}IDIA​
 presented below, investments are compounded over all calendar days in the respective interest period.
Consider a unit investment over a time period of dcd_cdc​
 calendar days. Let rir_iri​
 be an interest rate that is published once every business day and assume that the business day convention is such that the year has NNN
 days. We define an interest rate r~i\tilde{r}_ir~i​
 such that r~i\tilde{r}_ir~i​
 coincides with rir_iri​
 on business days and is set to the rate of the previous business day if iii
 is a holiday or a weekend. In this straightforward manner we obtain an interest rate for all calendar days and can now set
​IDIA=Ndc[∏j=1dc(1+r~jN)−1].I_{DIA}=\frac{N}{dc}\left[\prod_{j=1}^{d_c}\left( 1 + \frac{\tilde{r}_j}{N} \right) -1\right].IDIA​=dcN​[∏j=1dc​​(1+Nr~j​​)−1].
​
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